## Masters Theses

#### Abstract

"Data assimilation is a very powerful and efficient tool to use collected raw data for improving model prediction in numerical weather forecasting, hydrology, and many other areas of geosciences. In this thesis, an iterative algorithm [23] of variational data assimilation with finite element method is utilized to study different models. One motivation for this fundamental mathematical study is to provide a potential tool for simulation of CO2 sequestration by extending it to more realistic and sophisticated models in the future. The basic idea of variational data assimilation is to utilize the framework of optimal control problems. We apply the iterative algorithm with corresponding discretization formulation of the model equations to approximate the optimal control in the variational data assimilation problems. We conduct a group of comprehensive numerical experiments for both the second order parabolic equation and Stokes equation. These two models are critical to study Darcy's law and Stokes-Darcy problems for CO2 sequestration, especially for the CO2 storage in fractured reservoir and the leakage around the natural faults.

One key point for this method of data assimilation is the derivation of the adjoint models. For the convenience of computation, we discretize the adjoint models in the operator formulation into the corresponding discretized matrix formulation. We focus on the application of the iterative algorithm to the second order parabolic equation and Stokes equation with different numerical tests for the parameter sensitivity, convergence, accuracy, and efficiency of the algorithm.

At each step of the iteration, there are three major stages: solving original forward equation with the current control, solving backward adjoint equation with the observation and the current solution of the forward equation, and updating the control for the next iteration step. Finite elements are utilized for the spatial discretization, finite difference schemes are utilized for temporal discretization, and the conjugate gradient method is utilized for solving the control equation in order to update the control. The numerical results illustrate that the iterative algorithm is stable and efficient for variational data assimilation problems"--Abstract, page iii.

Zhang, Yanzhi

He, Xiaoming
Singler, John R.

#### Department(s)

Mathematics and Statistics

#### Degree Name

M.S. in Mathematics

United States. Department of Energy

#### Publisher

Missouri University of Science and Technology

Fall 2014

vii, 42 pages

#### Note about bibliography

Includes bibliographical references (pages 39-41).

#### Document Type

Thesis - Open Access

text

#### Language

English

Simulation methods
Mathematical models
Carbon sequestration -- Mathematical models

T 10596

902736374

COinS